Biomedical Engineering Reference
In-Depth Information
longer before recording the data; the plot is presented as it is to illustrate the
general point that slow relaxation near the critical point makes it more difficult
to get clean data there. Second, the chaotic region above B 0.81 appears rather
sparsely filled. Again, this is partly because the runs from which data were gath-
ered only covered 40 cycles of the drive. For longer runs, the data for a given B
would form rather dense bands with some visible gaps. Experimental data simi-
lar to Figure 5, however, would constitute clear evidence of chaotic behavior.
The presence of a period-doubling route to chaos in a wide variety of sys-
tems, together with the recognition that simple bifurcations can be classified into
generic types, is very encouraging. It means that many of the features of nonlin-
ear dynamical systems are
universal
, that is, they are independent of the quanti-
tative details of a model. There are now several other routes to chaos that have
been characterized, including quasiperiodic attractors that finally give way to
chaos, and intermittent behavior in which long periods of nearly regular behav-
ior are interrupted by chaotic bursts (17,23). This type of universality allows
educated guesses about how to construct models that exhibit the features ob-
served in experiments.
A basic vocabulary of bifurcations and transitions to chaos is now well de-
veloped, and one's first inclination upon observing chaos in an experiment
should be to classify its onset as a particular known type. The known classifica-
tion scheme is not exhaustive, however, and there continue to be cases in which
theoretical understanding requires exploring the mathematics of new types of
transitions. This is particularly true in systems with very many variables or sys-
tems described by partial differential equations that lead to complex patterns.
5.
TWO TYPES OF COMPLEXITY
:
SPATIAL STRUCTURE
AND NETWORK STRUCTURE
Thus far the discussion has been limited to systems with only a few degrees
of freedom. The effects of nonlinearity become much more difficult to charac-
terize or predict when many degrees of freedom interact. The complexity of the
solutions can become overwhelming, in fact, and many fundamental mathemati-
cal questions about such systems remain open. Nevertheless, the language and
techniques of nonlinear dynamics are helpful in formulating fruitful questions
and reporting results.
There are two different ways in which a system can involve a large number
of degrees of freedom, both of which are commonly encountered in biomedi-
cine. First, a system can be spatially extended, consisting of a few variables that
take on different values at different spatial points. Though such systems may be
described by just a few PDEs, the solutions can involve spatial structures of ex-
ceedingly complex form. A steadily driven chemical reaction, for example, can
display ever-changing patterns of activity as spiral waves are continually formed
